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Statement

With $C$ the set of integers having a monochromatic representation $n=a_1+a_2$, $a_1\ne a_2$, under a $k$-partition of $\mathcal N=\{1,2,\ldots\}$ (display (2), p. 47; as on the Theorem 1 page):

Theorem 3 (p. 55): "If $k\le3$, then for any $k$-partition of $\mathcal N$ there are infinitely many squares in $C$."

The theorem is introduced on p. 54 by "Our result is not strong enough to obtain for arbitrary $k$ that $a_1+a_2=x^2$ has a monochromatic solution with $a_1\ne a_2$. However a simple argument leads to". So the paper settles the square question for two and three colors and states that its Theorem 1 is not strong enough to give it for an arbitrary number of colors.

Source. P. Erdős, A. Sárközy and V. T. Sós, On a conjecture of Roth and some related problems I, in Irregularities of Partitions (Springer, 1989), 47--59; Theorem 3, Lemma 2 and the proof on printed p. 55 (PDF p. 9 of the scan), the introductory sentence on printed p. 54 (PDF p. 8). The scan's text layer garbles "$k\le3$" as "$k<3$" and "$\mathcal N$" as "M"; the statement was read on the page image.

Read depth. Claims checked: the theorem, Lemma 2 and the sentence introducing them were read clause by clause on the page images of pp. 54--55; the half-page proof was read for its structure and not checked step by step.

Proof sketch

Lemma 2 (p. 55, called "simple (and well known)"): for each $\varepsilon>0$, infinitely many integers $n$ can be written as $x^2+y^2$ in at least three ways (indeed in arbitrarily many) with both $x^2$ and $y^2$ in the window $[\frac n2(1-\varepsilon),\frac n2(1+\varepsilon)]$.

Take such an $n$ and three representations $x_1^2+x_6^2=x_2^2+x_5^2=x_3^2+x_4^2$ with all $x_i^2$ in the window. The paper states that the linear system $u_1+u_2=x_1^2$, $u_3+u_4=x_6^2$, $u_2+u_3=x_2^2$, $u_1+u_4=x_5^2$, $u_1+u_3=x_3^2$, $u_2+u_4=x_4^2$ in $u_1,\ldots,u_4$ has a solution in distinct positive numbers (the six sums are the six pairs of four unknowns, and the three representations make the system consistent). With at most three classes, two of the four $u_i$ lie in the same class, and their sum is one of the six squares, which therefore has a monochromatic representation with distinct summands. Infinitely many $n$ give infinitely many such squares. The solvability of the system in distinct positive numbers is asserted, not written out, in the paper and was not checked here.

Dependencies

Lemma 2 of the paper (stated as well known; no proof or reference given).

Bears on

• Problem 439: the paper's partial result for the square question, two or three colors; the paper says its Theorem 1 does not give the case of arbitrarily many colors, which the problem asks about and which Khalfalah and Szemerédi settled later.